ON NAPIER S RULES. 335 



angle are by Lemma 1 the negative complements of (3) and (4), 

 that is, they are those sides diminished by a quadrant. As the 

 points of intersection of (3) and (4) with (1) are respectively 

 distant by quadrants from the two extremities of (4), it follows 

 that the segment they cut off between them is the supplement 

 of (4). Hence the sides of the triangle and the complements of 

 its angles and hypothenuse are severally equal to the negative 

 complements of the sides of the pentagon, taken in the same 

 order as the five parts of the triangle. The same demonstration 

 will of course apply to the triarlgle formed by any other side of 

 the pentagon and its two opposites, the only difference being in 

 the starting-point of the cycle. Hence whatever relation con- 

 nects any one part of a right-angled spherical triangle as such 

 with its two opposites connects every other part with its two 

 opposites. Q.E.D. 



SCHOLIUM. 



The general principle that the sines of the sides are as those 

 of the opposite angles of course gives a relation between a part 

 and its opposites. But I should prefer to begin from that be- 

 tween hypothenuse and the sides, proving that, as in p. 331, from 

 a right-angled tetrahedron, which might be made the foundation 

 of spherical, as the right-angled triangle is of plane trigono- 

 metry. 



The relation connecting adjacent parts will be most simply 

 got by considering three consecutive relations of those which 

 connect opposites, multiplying the extremes and dividing by the 

 means. 



The general theory of spherical polygons must certainly 

 lead to some curious general results, but of course they are out 

 of my reach. 



The preceding paper was not sent to the Quarterly Journal, 

 because it was pointed out to me by a friend that a similar 

 resuscitation of Napier's own conception of his rules and a 

 similar remark upon the phenomenon of the disappearance of this 

 conception from modern English books had been already made 

 in the Course of Mathematics used in the Royal Military College 

 Dr Button's Course of Mathematics, edited by the late T. S. 

 Davies*. 



* Tutor to Mr Ellis ; see Biographical Memoir. 



